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We develop several notions of multiplicity for linear factors ofmultivariable polynomials over different arithmetics (hyperfields). The keyexample is multiplicities over the hyperfield of signs, which encapsulates thearithmetic of $\mathbf{R}/\mathbf{R}_{>0}$. These multiplicities give usvarious upper and lower bounds on the number of linear factors with a givensign pattern in terms of the signs of the coefficients of the factoredpolynomial. Using resultants, we can transform a square system of polynomialsinto a single polynomial whose multiplicities give us bounds on the number ofpositive solutions to the system. In particular, we are able to re-derive thelower bound of Itenberg and Roy on any potential upper bound for the number ofsolutions to a system of equations with a given sign pattern. In addition, ourtechniques also explain a particular counterexample of Li and Wang to Itenbergand Roy's proposed upper bound.

Factoring multivariate polynomials over hyperfields and the multivariable Descartes' problem